I am wondering if there are standard naming conventions in general for confidence distributions or for the particular cases of the non-centrality parameters of (non-central) $t$- and $F$-distributions. For the unitiated, let random variable $X$ have CDF $F(x;\theta)$. The confidence distribution CDF for $\theta$ is just $G(\theta;x) = F(x;\theta)$. This is somehow 'dual' to the original CDF. For creating confidence intervals one typically uses the quantile function of the confidence distribution, here $G^{-1}(q;x)$. So, given data $x$, the interval $\left[G^{-1}(1-\alpha/2;x),G^{-1}(\alpha/2;x)\right]$ is a $1-\alpha$ CI for the unknown $\theta$.

Are there standard names for these things? 'dual-t'? 'dual-F'?

Also, are there guidelines in R for this kind of thing? For example, dt, pt, qt, rt are the density, distribution, quantile and generator for the (non-central) $t$-distribution. What should I call the same functions for the confidence distribution? dcdt, pcdt, qcdt, rcdt?

If anyone has more knowledge of the history of the $t$- and $F$-distributions, I would gladly name them for pioneering researchers. For example, perhaps the $t$-distribution CD should be name for Johnson and Welch?

(edited again for clarity, probably still lacking same. Just looking for the names for these things.)

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    $\begingroup$ I'm confused about several aspects of this question. First, presumably your "$F(\theta;x)$" is an abuse of notation to mean $F(x;\theta)$ (for otherwise you have merely switched the names of the variables). But why can you call $F(x;\theta)$ qua function of $\theta$ a "distribution" when it is usually not normalized (and sometimes not even integrable)? Second, what is meant by $F^{-1}(\theta; x)$? If you are fixing $x$ and viewing this solely as a function of $\theta$, it's certainly not a "quantile function"! Third, what's the matter with the usual terminology, "likelihood"? $\endgroup$ – whuber Apr 20 '12 at 17:47
  • $\begingroup$ @whuber, sorry, it is still early here. Briefly, yes, the CDF of the CD on $\theta$ is $G(\theta;x) = F(x;\theta).$ I can see how this caused some confusion. By definition, if $x$ is drawn according to the parent distribution with true parameter $\theta$, then $G(\theta;x)$ is supposed to be uniform on $\[0,1\]$. I am by no means an expert on CDs, and would only be using the 'quantile' function for convenience. $\endgroup$ – shabbychef Apr 20 '12 at 18:33
  • $\begingroup$ I believe using mathematical concepts that represent a completely different idea, such as dual, is not very useful. $\endgroup$ – user10525 Apr 20 '12 at 18:40
  • $\begingroup$ @Procrastinator thanks for the input; the term 'dual' is also used in e.g. optimization, without confusion over the 'completely different' idea of dual vector spaces. But maybe 'co-t' and 'co-F' would be better, though? I value both terseness and unambiguity here. thanks. $\endgroup$ – shabbychef Apr 20 '12 at 18:49
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    $\begingroup$ ach, yes I edited the post, but forgot to make it $G^{-1}$. So, yes $G^{-1}(q;x)$ is a $1-q$ upper confidence limit for $\theta$ given data $x$. This is all based on the 'modern definition' of CD: en.wikipedia.org/wiki/…, which may be a bit, uh, obscure or weird. $\endgroup$ – shabbychef Apr 20 '12 at 19:20

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